1
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that
$$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F'\left( x \right) < 0$$ for all $$x \in \left( {{1 \over 2},3} \right).$$ Let $$f\left( x \right) = xF\left( x \right)$$ for all $$x \in R.$$

The correct statement(s) is (are)

A
$$f'\left( 1 \right) < 0$$
B
$$f\left( 2 \right) < 0$$
C
$$f'\left( x \right) \ne 0$$ for any $$x \in \left( {1,3} \right)$$
D
$$f'\left( x \right) = 0$$ for some $$x \in \left( {1,3} \right)$$
2
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right) = \int\limits_{{1 \over x}}^x {{e^{ - \left( {t + {1 \over t}} \right){{dt} \over t}}}} .$$ Then
A
$$f(x)$$ is monotonically increasing on $$\left[ {1,\infty } \right)$$
B
$$f(x)$$ is monotonically decreasing on $$(0,1)$$
C
$$f(x)$$ $$ + f\left( {{1 \over x}} \right) = 0$$, for all $$x \in \left( {0,\infty } \right)$$
D
$$f\left( {{2^x}} \right)$$ is an odd function of $$x$$ on $$R$$
3
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Let a $$\in$$ R and f : R $$\to$$ R be given by f(x) = x5 $$-$$ 5x + a. Then,
A
f(x) has three real roots , if a > 4
B
f(x) has only one real root, if a > 4
C
f(x) has three real roots, if a < $$-$$4
D
f(x) has three real roots, if $$-$$4 < a < 4
4
IIT-JEE 2012 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then
A
$$S \ge {1 \over e}$$
B
$$S \ge 1 - {1 \over e}$$
C
$$S \le {1 \over 4}\left( {1 + {1 \over {\sqrt e }}} \right)$$
D
$$S \le {1 \over {\sqrt 2 }} + {1 \over {\sqrt e }}\left( {1 - {1 \over {\sqrt 2 }}} \right)$$
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